I colour coded some parts of the sentence so hopefully you can see how it works. Any further questions? I think most of it is really just a case of fitting together the component parts correctly (e.g. knowing that lýsa yfir is one 'unit').

I can also see that reynist might cause confusion. If you see verbs like this in the subjunctive it can mean "if verb". In the case of reynast the subjunctive is unfortunately the same as the indicative in this instance, but you can tell it's the subjunctive from the sentence itself once you've had enough experience with the form. For example:

Búðin er lokuð. The shop is closed.Sé búðin lokuð þá verðum við að fara heim. If the shop is closed then we'll have to go home.

It's not easy to tell what you mean by "the harder things" so I am just guessing.

So, you should be able to make sense of this now: Sambandið hefur lýst því yfir að (The EU [lit. union] has declared that)
Then we get to a conditional sentence, which is what I imagine is throwing you off. There are normal ways to using conditionals with conjunctions like 'ef' and then you invert the subject and the verb. I know you've posted about conditional sentences here before so I know you're aware of them. I hinted at this structure in the last big paragraph in that thread (in post #4). What happens is, you can use the subjunctive to start off both clauses. So I'll bracket the clauses and highlight the verbs:

Then "gríðarlegt bakslag fyrir Bandaríkin" is just a literal -> enormous backlash for the United States.

So, putting all that together (and changing it to sound better in English), you get:

The European Union has declared that should these accusations turn out to be true, this would bring about an enormous backlash for the United States.

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I'd actually want to be more loose in the translation and say "this would bring out serious consequences for the U.S." but I thought it'd best to stick to a more literal/transparent translation since we're talking about what's going on in the sentence.

Ah, cross-posted with SB.
Although we chose different words, there isn't a 1-1 correspondence with a lot of terms and it's a matter of choice, but they are basically different words that mean exactly the same thing. "Prove to be" does sound better than "turn out" in a news context so that would be a better choice.

Totally didn't twig that Sambandið would be the EU in this context, which I suppose shows how well up I am on current affairs. I really enjoyed our tandem translations Just goes to show how much it is an art and not a science.

So what is it specifically that implies the "IF". I know the sentence woudlnt make sense without it but is there a particular thing which causes it to be. Also where Alxmrphi has placed squared brackets around the two clauses, is this then parellel to how commas work in English? Sort of if there is a switch such as það with muni this means there was an adverb or clause before it which causes there in english to be a comma? Just a random observation could be very loose.

Also I was wondering, if actually possible, if there could be a breakdown of the conjunctions themselves. An explanation as to why lýsa yfir means declare and why hafa í för með sér means what it does, what is the breakdown of all the words?

Well lýsa yfir technically would mean "illuminate over" or perhaps less literally "to shine light on", but it's not particularly helpful think of it like that (because that is not what it means!). Lýsa, related to the noun ljós as you can probably see, literally means "to light up", but is more commonly used to mean "describe" so although it is often good to understand literal meanings, you have to be careful. Just think that those two words together = declare. Hafa í för með sér is like "have in tow" "have with on a journey", just means that something accompanies something else.

It is the use of the subjunctive that denotes the if.

Eplið er rautt. The apple is red.Sé eplið rautt... If the apple is red...

About the commas... no, I would drop that if I were you. I think the use of a comma in the English translation would be a stylistic choice, and not grammatically necessary.

What implies the if is the fact that when you get to the first að the next thing that follows isn't usually a verb (because verbs go in the second slot in clauses) but if you get a verb in there and then another verb in the subjunctive, it implies the conditional. If you speak out the sentence to you with big overemphatic stress and breaking it up into the clauses, that often helps for me.

With the commas: sometimes we can use 'then' to join conditional clauses and in those cases you don't need a comma. The idea is similar though, yes. What is causing the shift is best not thought about as an adverb going before it, just the structure of this type of infrequently-used conditional.

Also I was wondering, if actually possible, if there could be a breakdown of the conjunctions themselves. An explanation as to why lýsa yfir means declare and why hafa í för með sér means what it does, what is the breakdown of all the words?

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You say 'conjunction' but I don't think it's what you mean. Conjunctions are words like 'and', 'if', 'then', 'so'.
It sounds like you just mean an explanation of the expressions. Is that right?

I've got no idea why lýsa yfir means declare, but hafa í för með sér is not totally random.
Something has something else with it on a journey. So it's obviously grammaticalised (i.e. not literal meaning but more metaphorical in a sense).
If something goes on a (here metaphorical) journey with something else, it's like 'bringing a consequence' with it. So the enormous backlash follows if the accusations are true.

Think about 'bring about' in English. It doesn't mean 'bring' as in its normal sense but it has the exact same sort of thing going on here.
An action of movement, implying consequence or cause. "Bring" shows one thing bringing another, exactly like 'having on a journey with' (i.e. bringing) in Icelandic.

English historically shares the exact same construction.
Albeit it's made it's way out of the language for the most part but we can still use it today (i.e. with the subjunctive clause-initial in the first clause).

Were I to win the lottery, I would buy you a car.
Were I in his shoes, I wouldn't have done that.

As you know the standard form to go with I is was. You put it into the 'subjunctive' were and can completely take out the if just like you can do in Icelandic.
Knowing that our native language has a form of exactly the same thing hopefully takes away some of the idea of a really different to comprehend grammatical structure.
Before we were just restricted to using wereandshould in Modern English, you could use it with main verbs as well (i.e. Won I the lottery...) but slowly English moved away from these. Icelandic still allows full verbs (i.e. non-auxiliaries) to do this in the construction and that's what is going on here.

So, i know off the original topic, and I know you have shown me the 4 uses involving subjunctive with... get, gat, gati og gæti

However the subjunctive in these examples were only used in larger constructions. Earlier I text my friend asking if they were free, they were not, so I then tried to say "I just wanted to catch up"... but I was thinking okay, it isn't past as I still WANT to im just expressing the want... I'm not sure how that would be categorized in english or I'm making something up but it didn't feel like past, as in, Yeh I wanted to yesterday but now I don't... so I thought maybe "ég vilji að hitta" was correct rather than vildi?

Well, you don't need the subjunctive at all in that case. You're expressing a genuine description of something truthful that happened (it is true that you wanted to do this).
That's why the indicative is fine. Now in the case where you talk about yourself, you use an infinitive that follows (and therefore it's non-finite and can't be conjugated into anything) so that's just a bare form. But in the cases where you would talk about someone else, then you're describing an action you wanted (so the verb of the action explaining this is factual and therefore the indicative mood is used) but when you go on to say what you wanted someone else to do, the moment has passed and they didn't actually do it so you're talking about a past action that you wanted to happen but didn't.

Ég vildi hitta (infinitive)
Ég vildi að hann hitti ... (subjunctive)

Weak verbs (except the -ja class) all have the same forms in the past tense between indicative and subjunctive so you can't see it here as being distinctive.
Remember that after vilja you just use the infinitive and no infinitive marker.

There you can see the distinction better. So, in the final example you were trying to make vilja the subjunctive because it's to do with wanting, but actually the subjunctive goes onto the verb that vilja introduces. Here, though, it's just the plain invariable infinitive so you don't have a subjunctive (but the subject and the verb aren't the same, then you see it - as in the last example I wrote).

Do english words (where possible) have subjunctive forms or is the subjunctive formed by a mannor of things (im starting to feel its more just the feeling of the sentence)

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The subjunctive is rarely used in English as I understand it, and subjunctive forms only differ from indicative forms for the verb to be, I think. It may be that because the forms are the same for other verbs, we native English speakers have lost the sense of how the subjunctive differs from the indicative - something we can only understand from the study of other languages like French and Icelandic that do use the subjunctive mood. I'm comfortable with using the subjunctive forms of to be (in constructions like: If I were a rich man) and using should in a subjunctive way (Should it come to pass), but I know I never even heard the word subjunctive before I was about 12 in French lessons at school. Nevertheless, I would not say that the subjunctive is "the feeling of the sentence" (although you will come to get a feeling for when it is appropriate), it is a grammatical mood indicated in languages that use it by a different way of conjugating a verb.

Ég vildi bjóða honum út að borða.
Ég bauð honum út að borða.
Ég vildi að hún myndi bjóða honum út að borða. (This is the only one that uses the subjunctive, twice because myndi is also a subjunctive form of munu. You could also probably say "Ég vildi að hún byði honum út að borða", where bjóða is in the subjunctive instead of using myndi.)

Ég vildi að hún myndi bjóða honum út að borða. (This is the only one that uses the subjunctive, twice because myndi is also a subjunctive form of munu. You could also probably say "Ég vildi að hún byði honum út að borða", where bjóða is in the subjunctive instead of using myndi.)

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What do you mean with the 'twice' here? Just so all is clear .

And a note to Jake: People often use this 'myndi + infinitive' with weak verbs because there is no distinction in past tense of the vast majority of weak verbs, this is a useful way to state that something is the subjunctive. This usage has a tendency to be overused and it's not really considered (in the books on style) good to use when the verb class actually does make a distinction between past indicative and past subjunctive (i.e. strong verbs). "Normal" Icelandic however does frequently have this myndi version with strong verbs but just be careful if you want to be taken more seriously later on/academically. I always try to do that but the subjunctive vowel shifts always drop out of my memory so myndi + infinitive is just a very handy exit strategy in that case. Overdoing it would probably sound a bit pompous. I'm sure once you start noticing it you'll easily place it within the different contexts you observe it and see how it is used.

Hmm OK as I understood it, you can use the past subjunctive to have a sort of "I would" effect and one of the verbs very commonly used this way is vilja.

Ég vil að hún komi - I want her to come
Ég vildi að hún kæmi - I wish she would come / I would like her to come

The same as:

Ég er til í það - I am up for that
Ég væri til í það - I would be up for that

I suppose there would be another option with the second example sentence since the past tense forms of vilja are the same for indicative and subjunctive:

Ég var til í það - I was up for that
Ég vildi að hún kæmi - I wanted her to come

But this is not what I meant with "Ég vildi að hún byði honum út að borða". Then vildi means not wanted but wish / would like, and that's the meaning that OP asked for... This is the reason I assume that vildi is the subjunctive in sentences like this one.

I don't think they mean I wanted but rather I wish / I would like. From my experience "Ég vildi að" is used all the time where in English we would say "I wish that". Hence it must be the subjunctive mustn't it? The more I think about it the more certain I am...

I wish I were a bird.
I wish I could dance.
I wish you were here.
I wish this was just a dream, I wish that my grandmother wasn't dead.

No, no I don't think you've got anything mixed up really. I think the only thing is the 'vildi' being subjunctive in the first clause. I'm pretty sure it's indicative when introducing it and the subjunctive comes about as a consequence of that statement. Is it not the same in French? It's like that in Italian. There are only a few instances I can think of with other verbs that do show a difference. Okay - biðja is a good example.

The meaning with vildi can definitely be more 'wish' but it's still a form of 'wanted' in the way I view it. It's less of a demand and thus expressed by 'wish' because of the fact it is the past and is often used like that, but you'd still translate 'wanted' as the same. I think we make a distinction that in Icelandic is subsumed by the same translation (as happens in other situations) and also like many times where Icelandic makes distinctions we don't. The logic on how it's interpreted is something I agree with you on, but that is more a consequence rather than a cause, if you get what I mean. The fact is saying that you wish something isn't enough to trigger the subjunctive, the wishing is a fact. If you wish something, even in the past, that is still true but what you wished for is the imagined non-true thing that is expressed in the subjunctive mood. Like segja, speaking is an action while what is then said is then reported and is indicative in both present and past tenses in the first clause, and the tense of the second clause follows from the first.

Just like the other verbs that fit in this group that are to do with demands and orders etc - the actions themselves then express non-true desires/orders/wishes and it's those things (in the following dependent clause) that take the subjunctive, not the ones that introduce them.

Then as a separate thing you can use a past conditional with a present meaning (i.e. conditional for politeness or diminished responsibility).
Like "I would be up for that" which is a past form used in a conditional manner and works the same way in Icelandic.

Hmm OK as I understood it, you can use the past subjunctive to have a sort of "I would" effect and one of the verbs very commonly used this way is vilja.

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Aha, I didn't see your edit. I see what you were thinking about now.
When you have the past tense (I'll refrain from saying 'indicative' here ) in the first clause then the past subjunctive in the second one is ambiguous in a lot of contexts between a past event or this "would" thing. I am actually sitting in a hospital with my book on Icelandic syntax in my bag (how sad am I). I'll quote the examples about this part

Jón sagði að þú tækir blöðin
- John said you took the papers
- John said you would take the papers

So these environments are ambiguous between the two interpretations and before when I first used the bjóða example I was thinking of the first one and you replied interpreting the second one. With vilja you could get readings either of wishing / wanting like in the second example here, or the one that I was referring to (the first) like "I wanted her to ask (i.e. that she asked) him out to dinner" rather than "I wanted (wished) that she would ask him out to dinner."

Funny how ambiguity can cause people to see things so differently!

Even if it was a direct wish it's in the clause introduced by the verb. Just think about að óska einhverjum einhvers, it's wishing someone something in the first clause (either in present or past) with the subjunctive following. Though yes, in this example the past tenses are identical, but my mind isn't coming up with a better example. With segja / krefjast and biðja though, they all follow that same pattern.

I would understand Jón bað að henni yrði vísað inn to mean that Jón asked for her to be directed in, which I agree is just the indicative past tense. Hann krafðist að Silver drægju tilkynninguna tilbaka, same story, he demanded, just simply the past tense. On the other hand I can't help seeing a difference between vildi-wanted and vildi-would want, it seems to me like the difference between var-was and væri-would be or hafði-had and hefði-would have. Why does "I would want" have to be true (since I don't think this is actually grammatically equivalent to "I wish")? I'm sure you're right, you have a much deeper technical understanding of language than I do, but I still don't really get it.
For interest's sake, why do these sentences have such different meanings?

Ég vildi vera strákur - I wanted to be a boy (a past desire, speaker no longer wants to be a boy)
Ég vildi að ég væri strákur - Lit. I wanted that I were a boy, in line with what you say, lit. I would want that I were a boy in line with what I was thinking (effectively I wish I were a boy, present and continuing desire)

I'm probably just muddling myself. It's a rather academic point in any case!